Loughborough UniversityInstitutional Repository

Kinetic models of micellesformation

This item was submitted to Loughborough University's Institutional Repositoryby the/an author.

Citation: STAROV, V., ZHDANOV, V. and KOVALCHUK, N.M., 2010. Ki-netic models of micelles formation. Colloids and Surfaces A: Physicochemicaland Engineering Aspects, 354 (1-3), pp.268-278

Additional Information:

• This article was published in the journal, Colloids and Surfaces A: Physic-ochemical and Engineering Aspects [ c© Elsevier]. The definitive version isavailable from: www.elsevier.com/locate/colsurfa

Metadata Record: https://dspace.lboro.ac.uk/2134/5839

Version: Accepted for publication

Publisher: c© Elsevier

Please cite the published version.

This item was submitted to Loughborough’s Institutional Repository (https://dspace.lboro.ac.uk/) by the author and is made available under the

following Creative Commons Licence conditions.

For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/

1

KINETIC MODELS OF MICELLES FORMATION

V. Starov1f ,V. Zhdanov2, N.M.Kovalchuk1,3

1Department of Chemical Engineering, Loughborough University, Loughborough,

LE11 3TU (UK) 2Moscow State University of Food Production, 11 Volokolamskoe sh., Moscow,

125080, Russia 3Institute of Biocolloid Chemistry, 03142 Kiev, Ukraine

Abstract

Four possible aggregation models in surfactant solutions are considered. It is shown

that only the model taking into account interactions between clusters of sub-micellar

size shows a transition to the micelles formation at a concentration above the CMC.

Key words: surfactant solution, cluster, micelle formation.

f Corresponding author, V.M.Starov@lboro.ac.uk

2

Introduction

Surfactants are widespread in nature, industry and everyday life [1-5]. They

play an important role in many technological applications, such as dispersion

stabilization, enhanced oil recovery, and lubrication. It may be argued that surfactants

are the most widely spread chemicals in the world.

Surfactant molecules are diphilic, with a hydrophilic head and a hydrophobic

tail. That is why they preferably adsorb on interfaces. They are soluble both in oil and

aqueous phase with solubility depending on their hydrophile-lipophile balance (HLB)

[6]. At low concentrations surfactant molecules are believed to exist in the solution

mainly as single molecules. If the concentration increases and reaches some critical

value, CMC, the surfactant molecules form new objects referred to as micelles [6-13].

In aqueous solutions hydrophobic tails are collected inside the micelle and only

hydrophilic heads are exposed to the aqueous phase.

In spite of the clear understanding of thermodynamic background of the

micelles formation [7,14], there is no kinetic theory at present, which can predict both

cluster formation (doublets, triplets and so on) below CMC and transition to the

micelle formation above the CMC in surfactant solutions based on their

aggregation/disaggregation rates.

Aggregation and disaggregation of single molecules and clusters of surfactant

molecules is a complex phenomenon, which is still to be understood. The theory of

aggregation (coagulation) of colloids was proposed by Smoluchowsky [15] and

further developed in [16], where disaggregation of colloids was introduced.

Application of such approach to surfactant solutions is referred to as a quasi-chemical

approach [13].

Theoretical models have been suggested, which allow evaluation of the

relaxation times associated with micellar solutions. A two-state model [17-18]

considers a monomeric state and an associated state consisting of all species larger

than the monomer unit. This model describes only the fast process (temperature-jump,

pressure-jump, stopped flow) and makes the assumption that the rate constant for

association and dissociation of the monomer from the micelle is independent of the

size of micelles. A theory of relaxation applicable for both slow and fast processes has

been developed [19-21] using a quasi-chemical approach. However, transition process

from monomolecular to micellar state in surfactant solutions was not considered in

these publications: it was taken for granted that the micelles formation already took

3

place. It was a reason why the value of CMC has not been determined in [19-21]

based on the aggregation/disaggregation model adopted in [19-21].

The aim of this paper is to establish the aggregation model, which predicts the

formation of clusters (doublets, triplets and so on) in non-ionic aqueous surfactant

solutions below the CMC and micelles formation above the CMC. The quasi-

chemical approach is used below.

In this part we briefly summarize the known theoretical results relevant to the

quasi-chemical approach of the micelles formation [13], [16], [22]. The terminology

used in [16], [22] is adjusted below for the consideration of surfactant solutions.

Let ni(t), i=1,2,3,…be the number concentration of clusters with 1,2,3, …

initial molecules at the moment t. The rate of aggregation of two clusters of sizes i

and j in one bigger cluster of size i+j is jiji nna , , where ai,j, i, j =1,2,3,… are

corresponding aggregation rates. The rate of disaggregation of the cluster of size i+j

into two smaller clusters of sizes i and j, respectively, is jiji nb +, , where bi,j, i, j

=1,2,3,…are disaggregation rates. Aggregation/disaggregation rates satisfy the

following symmetry conditions: ai,j=aj,i, i, j =1,2,3,…, bi,j=bj,i, i, j =1,2,3,….

Using the above notations development over time of cluster concentrations can

be written as [16], [22]:

,...3,2,1,21 1

1 1,, =Ψ−Ψ= ∑ ∑

−

=

∞

=− k

dtdn k

i iikiki

k , (1)

where jijijijiij nbnna +−=Ψ ,, .

The first sum in the right hand side of Eq. (1) represents all

aggregation/disaggregation events with cluster those sizes range from 1 to k-1 (the

total flux to the state k from all possible states i<k), while the second sum in the right

hand side represents all aggregation/disaggregation events with clusters those sizes

range from k to ∞ (the total flux from the state k to all states i>k).

System of differential equations (1) can be rewritten in a more conventional

form as

,...3,2,1,22

1 1

1 1,

1

1 1,,, =+−−= ∑ ∑∑ ∑

−

=

∞

=+

−

=

∞

=−−− knbnanbnnna

dtdn k

i iikik

k

i iiikkiki

kikiiki

k . (2)

It is possible to show that the latter system of differential equations satisfies

the condition of conservation of the total number of surfactant molecules in the

4

system under consideration at any aggregation/disaggregation rates ai,j, i, j =1,2,3,…,

bi,j, i, j =1,2,3,…, which satisfy the symmetry conditions:

Nknk

k =∑∞

=1, (3)

where N is the initial number concentration of single surfactant molecules.

Let ∑∞

=

=1k

knn be the total number of aggregates. Using Eq. (2) it is possible to

conclude that

( )∑∞

=+−−=

1,,,2

1ji

jijijiji nbnnadtdn . (4)

Let us consider the steady state solution of the system of Eqs. (2), that is,

,...3,2,1,22

101

1 1,

1

1 1,,, =+−−= ∑ ∑∑ ∑

−

=

∞

=+

−

=

∞

=−−− knbnanbnnna

k

i iikik

k

i iiikkiki

kikiiki (5)

The important conclusion obtained in [13], [22] is as follows: the steady state

solution of the system (5) corresponds to the minimum of the free energy of the

system under consideration.

Models of aggregation/disaggregation

Below we distinguish between clusters (doublets, triplets, and so on, with

number of surfactant molecules smaller than in micelles) and micelles itself. Four

different aggregation/disaggregation models are considered below. It is shown that

only one of these models, Model C, results in a transition from low sized cluster

formation to the micelles formation at and above some critical concentration of

surfactant molecules. All other models show a continuous increase in averaged cluster

size with the increase of the surfactant concentration.

Model A (Fig. 1, A1 and A2): aggregation/disaggregation of surfactant

molecules according to this model occurs via exchange by one molecule at the time

between clusters/micelles as shown in Fig. 1 (A1 and A2) there only single molecules

can be connected/disconnected to/from any cluster (including micelles if any). This

model corresponds to that proposed in [19-21] and generally accepted now.

According to Model B (Fig. 1, B1 and B2), aggregation/disaggregation of

clusters of any size can take place.

5

Connection/disconnection of clusters/individual surfactant molecules go in a

symmetrical way according to Models A and B.

With Model C aggregation/disaggregation of clusters (or micelles if any)

occurs asymmetrically: clusters of any size can aggregate but only single molecules

can leave clusters or micelles. Fig. 1 (C1 and C2) shows that clusters of different sizes

can be connected into a new bigger cluster/micelle but only single molecules can

disconnect from the cluster/micelle.

Usually Models B and C are excluded from the consideration arguing that

there is a strong bimodal distribution (single molecules and equilibrium micelles) in

surfactant solutions above the CMC, concentration of submicellar clusters is small

and therefore contribution of cluster/cluster interaction could be neglected. It is true

for solutions close to equilibrium at concentrations far above the CMC. However, we

consider the equilibration process, which started from the solution, where only

monomers are present. Therefore, presence of small clusters is inevitable and should

be taken into account. Moreover, even in the solutions close to equilibrium at

concentrations close to CMC the contribution from the small clusters in the relaxation

processes is sometimes very important [23,24].

With Model D aggregation/disaggregation of clusters (or micelles if any)

occurs also asymmetrically: only single molecules can join clusters but clusters of any

size can disaggregate. Fig. 1 (D1 and D2) presents this situation.

Being aware that the probability of realization either Model B or D is rather

small, because in this case several intermolecular bonds should be broken

simultaneously, we still consider these Models for the sake of completeness.

Analytical solutions and numerical simulations below show, that in the case of

Models A, B, and D equilibrium distribution of doublets, triplets and so on develops

continuously with concentration and does not undergo a transition to the micelles

formation at any concentration.

Situation is completely different (see below) in the case of the Model C (Fig.

1, C1 and C2): equilibrium distribution of low sized clusters (doublets, triplets, and so

on) is possible only at concentrations below some critical. Above this critical

concentration the system undergoes a transition to the micelles formation which

results in the formation of a new very distinct bimodal distribution. The latter means

that this critical concentration is the CMC.

6

In the case of pure Brownian aggregation jia , , i,j=1,2,3,…are determined by

Smoluchowsky [15,25] as

( )jiji

ji aaaa

kTa +⎟⎟⎠

⎞⎜⎜⎝

⎛+=

1132

, μ, (6)

where k is the Boltzmann constant, T is the absolute temperature, μ is the dynamic

viscosity of dispersion medium or solvent in the case of surfactant solutions.

Following Smoluchowsky [15,25] it is assumed further that in Models B and C

collisions occurs mainly between particles of close sizes and therefore for these

models ai,j=aB,C=8kT/3μ. Obviously for Models A and D collision occurs mainly

between particles of different size. It was assumed that in this case ai,j=const=aA>aBC.

Disaggregation rates jib , , i,j=1,2,3,…depend solely on the interaction energy

between clusters and these coefficients are also assumed independent of the cluster

size. That is, in all four models below aggregation/disaggregation rates are assumed to

be constant, aI and bI, respectively, independently of the cluster size.

Model A.

According to this model only single molecules can connect/disconnect to/from

clusters.

All possible events with a cluster of size k, k=1,2,3,4,… are as follows:

• connection of one molecule to a cluster of k-1 size, which results in an increase of

nk value; the reaction rate of this process is aA. However, at k=2 the reaction rate

is 2aA ;

• disconnection of one molecule from a cluster of size k+1, which results in an

increase of nk value; the reaction rate of this process is bA or 2bA at k=1;

• disconnection of one molecule from a cluster of size k, which results in a decrease

of nk value; the reaction rate of this process is bA;

• connection of one molecule to a cluster of size k, which results in a decrease of nk

value; the reaction rate of this process is aA or 2aA at k=1.

Taking all these events into consideration the following system of equations

can be deduced from the general system (2)

7

,...4,3,2,

)(

1111

21211

1

=−−+=

+−+−−=

+− knnanbnbnnadt

dn

nbnnbnannadtdn

kAkAkAkAk

AAAA

(7)

with conservation condition of conservation of the total number of particles (3).

Under steady state conditions the left hand sides of Eqs. (7) should be set to

zero. Let fk=nk/N, k=1,2,3, … be the fraction of clusters of size k, and α=aAN/bA be

the dimensionless concentration. Using these notations system of Eqs. (7, 3) under

steady state conditions takes the following form

11110 ffffff kkkk αα −−+= +− , k=2,3,… (8)

11

=∑∞

=k

kfk (9)

Solution to system of Eqs. (8), (9) is deduced in Appendix 1 (Eqs. (A1.10)-(A1.11)):

25.05.01

1+++

=αα

f (10)

k

k

kf]25.05.0[

1

+++=

−

αα

α, k=2,3,… (11)

It is possible to conclude using Eq. (11) that ,...4,3,2),( =kfk α dependencies go

from zero at α=0 to zero at α→∞ via the maximum value (see Appendix 1 for details)

at

,...4,3,2,4

12

max, =−

= kkkα (12)

and that maximum value is equal to

,...4,3,2,)1()1(4 1

1

max, =+−

= +

−

kikf k

k

k (13)

Dependencies fk(α), k=1, 2, 3, … according to Eqs. (10, 11) are shown in Fig.

2. Note, α is the dimensionless concentration. Fig. 2 shows that there is no restriction

on concentration in the Model A. Let us introduce ∑=∞

=1)(

kkfF α . It is easy to see that

the average cluster size, <k>=1/F(α). Using Eqs. (8)-(9) we can conclude that the

averaged cluster size in the case under consideration is 25.05.0 ++>=< αk . That

8

is, <k> is an increasing, convex function of the dimensionless concentration, α, and

this dependency does not have any inflection point.

That is, there is no CMC and there is no transition to the micelles

formation in the Model A.

Model B.

All possible events with a cluster of size k are as follows (Fig. 1, B1 and B2):

• connection of one cluster of size i to a cluster of k-i size (i=1,2,…k-1), which

results in an increase of nk value; the reaction rate of this process is aB;

• disconnection of a cluster of size k from a higher cluster, k+i, i=1,2,…, which

results in an increasing of nk value; the reaction rate of this process is bB. If i=k

then this reaction results in a formation of 2 clusters of k size, this means this

reaction rate is 2bB in this case;

• disconnection of cluster of any size from the cluster k size, i=1,2,.., k-1; reaction

rate is bB. However, at i=k/2 the reaction rate is 2bB. Hence, the cases of even and

odd k should be considered separately;

• connection of clusters of size k and i and i=1,2,… reaction rate is a. If clusters are

of the same size k, then reaction rate is 2aB.

Eq. (2) now can be rewritten as

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+−+−−−=

+−+−−−+=

+−+−−=

+

+

=

∞

=+

∞

=++

=−+

+

=

∞

=

∞

=

−

=−

∞

=

∞

=

∑∑∑∑

∑∑∑∑

∑∑

24

12

11

212

11212

2

112

12

4

2

11

22

122

212

12

2

211

21

11

1

2

22

kB

k

iiB

iiBkB

iikBkB

k

iiki

Bk

kB

k

iiB

iiBkB

iikBkBk

Bk

iiki

Bk

BBi

iBBi

iB

nbnbnbnannankbnnadt

dn

nbnbnbnannankbnannadt

dn

nbnbnbnanandtdn

k=1,2,3,… (14)

with the conservation condition of the total number of particles (3) satisfied.

Under steady state conditions and using fraction of clusters of size k, fk=nk/N,

k=1,2,3, … and the dimensionless concentration, α=aBN/bB, as in the Model A, the

latter system becomes:

9

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+−+−−−=

+−+−−−+=

+−+−−=

+

+

=+++

=−+

=

−

=−

∑∑

∑∑

24

12

1

2121212

2

112

4

2

1

2222

212

12

212

11

20

220

0

k

k

iikkk

k

iiki

k

k

iikkkk

k

iiki

ffffffkfff

ffffffkffff

ffffff

ααα

αααααα

k=1,2,3,… (15)

with the conservation law (9).

The system of Eqs. (15), (9) has exactly the same solution as the Model A,

which is given by Eqs. (10), (11). The latter can be checked by the direct substitution

of Eqs. (10), (11) into system of Eqs. (15), (9).

The latter means that there is no restriction on concentration in Model B. That

is, there is no CMC and there is no micelles formation according to both Model B

and Model A.

Model C.

According to this model any two clusters of different sizes can be connected in

a new bigger cluster but only single molecules can leave clusters (Fig. 1, C1 and C2).

All possible events with a cluster of size k, k=1,2,… are as follows:

• connection of one cluster of size k-i to a cluster of i size (i=1,2,…, k-1), which

results in an increase of nk value; the reaction rate of this process is aC=aB;

• disconnection of one molecule from a cluster of size k+1, which results in an

increase of nk value; the reaction rate of this process is bC=bA. Disconnection of

one molecule from any doublet results in a creation of 2 single molecules, this

means, the reaction rate of this process is 2bC;

• connection of a cluster of size k to any other cluster of i size (i=1,2,3,…), which

results in a decrease of nk value; the reaction rate of this process is aC. If i=k then

the reaction rate is 2aC;

• disconnection of one molecule from a cluster of size k, which results in a decrease

of nk value; the reaction rate of this process is bC.

System (2) in this case transforms into

10

⎪⎪⎩

⎪⎪⎨

⎧

>+−−−−+

+=

++−−=

∑ ∑

∑∑−

=

∞

=+−

∞

=

∞

=

1

1 11

22

22

21

11

1

1,4

)1(121

,

k

i ikCkCikCkCk

k

CikiCk

Ci

iCCi

iC

knbnannanbnannadt

dn

nbnbnannadtdn

(16)

Under steady state condition system (14) can be rewritten as

( )

⎪⎪⎩

⎪⎪⎨

⎧

>+−−

−−−=

+−+−−=

∑

∑−

=+−

∞

=

1,4

)1(3210

,0

1

11

2

2121

11

knbnannanbnna

nbnnbnanna

k

ikCkC

k

kCkCikiC

CCCi

iC

, (16’)

where ∑∞

=

=1i

inn is the total number of clusters including single molecules (clusters of

size 1). Using dimensionless concentrations introduced as

,...3,2,1,2/2/ === kfbaNnz kCCkk α the latter system of equations and the

conservation law (3) can be rewritten as:

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

>+−−

−−−=

+−+−−=

∑

∑∞

=

−

=+−

2

1,2

)1(320

220

1

1

11

2

21211

αi

i

k

ikk

k

kkiki

iz

kzzzzzzz

zzzzzz

, (17)

where ∑∞

=

=1i

izz .

Below system of algebraic Eqs. (17) is simplified as follows: it is ignored that

the aggregation of two equal sized clusters results in a twice higher reaction rate, that

is, terms 221 2

)1(3,2 k

k

zz −− are omitted in first and second equations of the

system of Eqs. (17), respectively. This is done to make it possible to get an analytical

solution of Eqs. (17). Using direct numerical simulation of the system of Eqs. (16) we

show (see below) that our conclusions based on a simplified the system of Eqs. (16’)

remain valid.

Using these simplifications the system of Eqs. (17) becomes:

11

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

>+−−=

+−+−=

∑

∑∞

=

−

=+−

2

1,20

20

1

1

11

211

αi

i

k

ikkkiki

iz

kzzzzzz

zzzzz

(18)

Note, both systems of Eqs. (17) and (18) independently satisfy the

conservation condition of the total number of particles.

Second equations of the system of Eqs. (18) can be summarized over k from 2

to infinity, which results in

( ) ( ) ( ) 212

21112

2

1

11

22

20

zzzzzzzzzzzzz

zzzzzzk

k

ikkkiki

−+−=−−+−−−−

=⎟⎠⎞⎜

⎝⎛ +−−=∑ ∑

∞

=

−

=+− (19)

Using Eq. (19) the system of Eqs. (18) can be rewritten as

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

>−+=

−−=−=

∑

∑∞

=

−

=−+

2

1,2

2

1

1

11

112

21

αi

i

k

iikikkk

iz

kzzzzzz

zzzzzzzz

(20)

Let us try to find the solution of the system of Eqs. (20) in the following form

,...3,2,1,11 =+−= ++ kzSzSz k

kk

kk , (21)

where ,...3,2,1, =kSk are unknown coefficients. Substitution of expressions (21)

into first three equations of the system of Eqs. (20) shows, that (a) solution in the

suggested form (21) really exists and (b) unknown coefficients should satisfy the

following relation

,...4,3,2,1

1== ∑

−

=− kSSS

k

iikik (22)

with the initial condition:

11 =S , (23)

12

which follows from the first Eq. (20) (see Appendix 2 for details).

Let us multiply both sides of Eq. (22) by zk, k=1,2,3,.. . After summation over

k from 1 to infinity this gives:

( ) ( )( ) ( ) zzYzSzSzSzYk

k

i

ikik

ii

k

kk +=== ∑∑∑

∞

=

−

=

−−

∞

=

2

1

1

11.

where ( ) ∑∞

=

=1k

kk zSzY . The latter equation can be rewritten as

( ) ( ) zzYzY += 2 (24)

If we take into account condition Y(0)=0, which follows from the definition of Y(z),

then the solution of Eq. (24) is

( ) zzY −−=41

21 (25)

This solution is defined only if

25.0≤z . (26)

Let us substitute solution in the form (21) into the last equation of the system

(20), which gives after some transformations (see Appendix 2): 2

)( α=zY , or, using

Eq. (25),

α−=− 141 z (27)

It is easy to see from the latter equation that solution exists only if 1≤α , or

C

C

abN ≤

If N>bC/aC then the equilibrium solution does not exist and formation of clusters of an

infinite size starts. That means,

CMCab

C

C = (28)

in this model.

That is, the Model C results in the existence of CMC, which is determined

by Eq. (28). Below CMC equilibrium clusters form: doublets, triplets and so on and

13

cluster size distribution is similar to that in the Models A and B. Above CMC system

under consideration can not be any more in equilibrium and formation of micelles

starts.

Distribution of fraction of clusters below CMC calculated according to Eqs.

(A2.6) is given in Fig. 3. The average cluster size, <k>, is as follows:

2/11

2 αα

−=>=<

zk . The latter dependency is an increasing but concave function at

concentrations below CMC. As soon as concentration reaches CMC, cluster size

changes from 2 to infinity. That is, CMC can be considered as an “inflection point”

and this gives an important hint for the subsequent consideration.

According to the consideration above the formation of micelles of an infinite

size starts above CMC. In our computer simulation below we introduce an

equilibrium number of individual surfactant molecules in a micelle, ℵ, which is used

as a parameter in our calculations below.

Model C: computer simulations.

Computer simulations are carried out to solve the kinetic Eqs. (2) in the case

of the Model C:

,..3,2,1,22

1

1 1,

1

1 1,,, =+−−= ∑ ∑∑ ∑

−

=

∞

=+

−

=

∞

=−−− kfBfAfBfffA

ddf k

i iikik

k

i iiikkiki

kikiiki

k αατ

(29)

where τ=tb is the dimensionless time, with initial conditions

,...4,3,2,0)0(,1)0(1 === iff i (30)

Coefficients ijA and ijB are selected as follows:

⎪⎩

⎪⎨

⎧

ℵ≥ℵ≥ℵ<=

≠ℵ<ℵ<=

,,,0,,,2

,,,,1

joriifiandjiif

jiandjiifAij (31)

and

⎪⎩

⎪⎨

⎧

>>==

≠===

.1,1,0,1,1,2

,,1,1,1

jandiifjandiif

jiandjoriifBij , (32)

14

where ℵ is the equilibrium number of individual surfactant molecules in a micelle.

The latter choice of coefficients means that any two clusters of size below ℵ can

aggregate, however, if the resulting cluster includes more than ℵ surfactant molecules

then this cluster is not equilibrium one and surfactant molecules can only leave this

cluster one at the time until the equilibrium number of molecules in the cluster, ℵ, is

reached.

Transient behavior and equilibrium solution (at t→∞) of Eqs. (29)-(32) are

discussed below. Solutions depend on both the dimensionless concentration of

surfactant molecules, α, and the equilibrium number of molecules in micelles, ℵ .

Transient behavior of dimensionless concentration of clusters is presented in

Fig. 4, calculated according to Eqs. (29)-(32) at the dimensionless concentration α=15

and the equilibrium number of individual molecules in micelles, ℵ=200.

Concentration α=15 according to our previous consideration should be well above the

CMC.

Fig. 4 shows the time evolution of cluster sizes according to Model C. After initial

lag time a bi-modal size distribution forms, which develops in time.

In Fig. 5 the equilibrium distribution of dimensionless concentration of clusters on

dimensionless concentration of surfactant molecules is presented calculated according

to Eqs. (29)-(32) at ℵ=200. Calculations are carried out at α=1.5 (close to the

dimensionless CMC), α=15 (the same as in Fig. 4), α=150. Concentrations of

micelles progressively increases with concentration, while the distribution of low

sized clusters remains almost unchanged.

In Figs. 6 and 7 dependences of averaged cluster size on dimensionless

concentration are presented for two cases ℵ=200 (Fig. 6) and ℵ=50 (Fig. 7). In both

figures these dependences have an inflection point between 1 and 1.5, which

corresponds to the CMC in dimensionless units.

Model D.

All possible events with a cluster of size k are as follows:

• connection of one molecule to a cluster of k-1 size, which results in an increase of

nk value; the reaction rate of this process is aD=aA;

15

• disconnection of a cluster of size k from a higher cluster, k+i, i=1,2,…, which

results in an increasing of nk value; the reaction rate of this process is bD=bB. If

i=k then this reaction rate is 2bD;

• disconnection of cluster of any size from the cluster k size; reaction rate is bD;

• connection of one molecule to a cluster of size k, which results in a decrease of nk

value; the reaction rate of this process is aD.

Eq. (2) now can be written as

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+−+−−=

+−+−−=

++−−−=

+

+

=++

+

=−

∑

∑

24

12

11121221

12

4

2

1122121

2

21121

1

kD

k

iiDDkDkDkD

k

kD

k

iiDDkDkDkD

k

DDDDD

nbnbnbnnankbnnadt

dn

nbnbnbnnankbnnadt

dn

nbnbnbnnanadtdn

k=1,2,3,…(33)

with the conservation condition of the total number of particles (3).

Using fraction of clusters of size k, fk=nk/N, k=1,2,3, …, dimensionless

concentration, α=aDN/bD and the dimensionless time tb=τ the latter system

becomes:

,...3,2,1

0)0(,

0)0(,

1)0(,

1224

12

11121221

12

24

2

1122121

2

12112

11

=

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=+−+−−=

=+−+−−=

=+−+−−=

++

+

=++

+

=−

∑

∑

k

ffffffkfffd

df

ffffffkfffddf

fffffffddf

kk

k

iikkk

k

kk

k

iikkk

k

αατ

αατ

αατ

(34)

with the conservation law (9).

Direct numerical solution of this system of differential equations was

undertaken. Only equilibrium solution of the latter system (that is solution at ∞→τ )

are discussed below.

Figs. 8 and 9 show results of our calculations. In Fig. 8 dimensionless

concentration of clusters is presented. This figure shows that according to the Model

D the concentration of clusters goes via its maximum value at n=2 (doublets),

16

concentration of doublets increases and the distribution becomes wider with

concentration. However, there is no transition to micelle formation at any

concentration.

In Fig. 9 dependence of the averaged cluster size on concentration is

presented. This dependence is the convex function and does not have any inflection

point in the whole range of concentrations. The latter observation confirms that there

is no transition to micelle formation according to the Model D.

Conclusions

Four possible models of cluster formation in surfactant solutions are

considered. It is shown that only one of these models shows a transition to the

micelles formation at concentration above some critical, which corresponds to CMC.

Three other models show a continuous increase in an averaged cluster size with

concentration and do not show transition to micelles formation.

Model A (Fig. 1, A1 and A2): aggregation/disaggregation of surfactant

molecules occurs via exchange by one molecule at the time between clusters/micelles.

Model B (Fig. 1, B1 and B2), aggregation/disaggregation of clusters of any

size can take place.

Connection/disconnection of clusters/individual surfactant molecules go in a

symmetrical way according to Models A and B. The equilibrium cluster

concentrations are identical in the case of these two models and do not show transition

to the micelles formation at any concentration.

With Model C (Fig. 1, C1 and C2) aggregation/disaggregation of clusters (or

micelles if any) occurs asymmetrically: clusters of any size can aggregate but only

single molecules can leave clusters or micelles.

With Model D (Fig. 1, D1 and D2) aggregation/disaggregation of clusters (or

micelles if any) occurs also asymmetrically: only single molecules can join clusters

but clusters of any size can disaggregate.

Solutions and numerical simulation below show, that in the case of Models A,

B, and D equilibrium distribution of doublets, triplets and so on develops continuously

with concentration and do not undergo transition to the micelles formation at any

concentration.

17

Situation is completely different in the case of the Model C (Fig. 1, C1 and

C2): equilibrium distribution of low sized clusters (doublets, triplets, and so on) is

possible only at concentrations below some critical. Above this critical concentration,

CMC, the system undergoes a transition to the micelles formation. If the surfactant

concentration is above the CMC then the system does not have an equilibrium

solution and shows a transient behavior, which results in the formation of a new very

distinct bimodal distribution: low sized clusters (single molecules, doublets, triplets

and so on) and micelles.

Acknowledgement

The authors would like to acknowledge the support from The Engineering and

Physical Sciences Research Council, UK (Grant EP/C528557/1) and by the EU under

Grant MULTIFLOW, FP7-ITN- 2008-214919.

18

Appendix 1

Model A. Solution of system of Eqs. (6,7).

Let us determine

∑∞

=

=1k

kff (A1.1)

Taking definition (A1.1) into account and

∑ ∑∑∞

=

∞

=−+

∞

=

=−−=−=2 2

12112

1 ,,k k

kkk

k fffffffff

after summation of all Eqs. (6) over k from 2 to infinity the following equation is

obtained

212 ff α= (A1.2)

From Eqs. (6) at k=2 one can conclude using Eq. (A1.2)

31

23 ff α= (A1.3)

From Eqs. (6) at k=3 one can conclude using Eq. (A1.3)

41

34 ff α= (A1.4)

From Eqs. (A1.2)-(A1.4) using Eqs. (6) it is possible to conclude that for any k=2,3,4,

...

...3,2,11 == − kff kk

k α (A1.5)

and the only unknown value is f1 , which should be found using Eqs. (7) and (A1.5).

Combination of these equations results in

11

11 =∑

∞

=

−

k

kk fkα (A1.6)

Differentiation of Eq. (A1.6) with respect to α shows that f1'(α)<0 that is f1(α) is a

decreasing function of dimensionless concentration α. It is easy to see that f1(0)=1

should be satisfied.

19

Solution of Eq. (A1.6). In order to solve Eq. (A1.6) the following function Ω(x) is

introduced 1

11',

1)(

−∞

=

∞

=∑∑ =Ω

−==Ω

k

k

k

kkx

xxxx

Using αf1 as x results in

1

11

11 1

)()(f

fff k

k αααα−

==Ω ∑∞

=

(A1.7)

and

11

1

1

1

11

1

11)('f

fkf

fk k

k

kk

k===Ω ∑∑

∞

=

−−∞

=

αα , (A1.8)

where at the last step Eq. (A1.6) is used.

It is easy to calculate the first derivative using Eq. (A1.7) in a different way as

21)1(

1'fα−

=Ω (A1.9)

Comparing Eqs. (A1.8) and (A1.9) results in the following equation for f1

determination

211 )1(

11ff α−

=

Solution to the latter equation, taking into account that f1=1 at α=0 results in

25.05.01

1 +++=

ααf (A1.10)

Now from Eqs. (A1.5) and (A1.10)

,...4,3,2,]25.05.0[

1

=+++

=−

kfk

k

k αα

α (A1.11)

The total number of particles and the averaged number of particles in clusters are

25.05.0 ++ αN

and 25.05.0 ++>=< αk , respectively.

From Eq.(A1.11) we conclude, that fk, k=2,3, ...dependencies on α go from

zero at α=0 via maximum value to zero at α→∞. Maximum values are reached at

20

,...3,2,4

12

max, =−

= kkkα (A1.12)

and that maximum value is equal to

,...3,2,)1()1(4)()(max 1

1

max, =+−

== +

−

kkkff k

k

kkk αα (A1.13)

21

Appendix 2

Model C. Solution of the system of Eqs. (18).

In this section we show that expressions (19)-(21) give the solution of the

system of Eqs. (18).

Let us compare

,...3,2,1,11 =+−= ++ kzSzSz k

kk

kk (A2.1)

at k=1 with the first equation of the system (18) 2

1 zzz −= (A2.2)

Comparison gives 11 =S and 11

12 ==∑

=iii SSS . Hence, expression (A2.1) and

,...4,3,2,1

1== ∑

−

=− kSSS

k

iikik (A2.3)

is valid at k=1 and the first equation of system (18) is satisfied.

Substitution of the Eq. (A2.1) into the second equation of system (18)

∑−

=−+ −+=

1

11 2

k

iikikkk zzzzzz (A2.4)

gives

.

221

1

1

1

11

1

1

11

1

1

211

121

11

11

22

∑∑∑∑−

=−

−

=

++−

−

=

+−+

−

=

++−+

+++

++

++

++

−++

−+−+−=+−k

i

kiki

k

i

kiki

k

i

kiki

k

i

kiki

kk

kk

kk

kk

kk

kk

zSSzSSzSSzSS

zSzSzSzSzSzS

or

0

222

1

1

1

11

1

111

1

11112

2

=⎥⎦

⎤⎢⎣

⎡−+

+⎥⎦

⎤⎢⎣

⎡−−−+⎥

⎦

⎤⎢⎣

⎡++−

∑

∑∑∑−

=−

−

=+−

−

=−++

−

=+−+++

k

iikik

k

iiki

k

iikikk

k

iikikk

SSS

SSSSSSzSSSSz

(A2.5)

The latter equation should be zero at any z, this means that all three expressions in

square brackets should be equal to zero. Expression in the third square brackets is

equal to zero according to the definition Eq. (A2.1).

The expression in the first square bracket of Eq. (A2.5) can be rewritten as:

22

02222

22

12121

1

1212

2212

1

11112

=−++−=−++−=

=++−=++−

+++++

+

=+−++

=+−++

−

=+−+++

∑

∑∑

kkkkk

k

jjkjkk

k

jjkjkk

k

iikikk

SSSSSSSSS

SSSSSSSS

The expression in the second square bracket of Eq. (A2.5) can be rewritten as

02222

2222

111

11

111

1

11

211

1

11

1

111

=−=+−+−−=

=−−−=−−−

∑∑∑

∑∑∑∑

=+−+

=+−

=+−+

−

=+−

=+−+

−

=+−

−

=−++

k

jjkjkk

k

iikik

k

jjkjkk

k

iiki

k

jjkjkk

k

iiki

k

iikikk

SSSSSSSSSSS

SSSSSSSSSSSS

according to Eq. (A2.3). This means that solution of system (18) is really given by

Eqs. (19)-(21).

Substitution of relations (A2.1) into the left hand side of the third equation of

system (18) gives

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) )(

)1(

121

2 122 1

1 1 1 1

11

11

zFzSzSzS

zSkzSzSkzSkzSk

zSkzSkzSzSkkz

k

kk

k

kk

k k

kk

k

kk

kk

k k

kk

kk

k k k k

kk

kk

kk

kkk

==+=

=++−=+−−=

=+−=+−=

∑∑

∑ ∑∑∑ ∑

∑ ∑ ∑ ∑

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

∞

=

++

++

From system (18) we can now find 43

332

22

1 52,2, zzzzzzzzz −=−=−=

or

ααα /)52(2,/)2(2,/)(2 433

322

21 zzfzzfzzf −=−=−= (A2.6)

where4

2 2αα −=z .

Distribution of volume fraction of clusters below CMC is given in Fig. 3.

23

REFERENCES

1. D.F. Evans, H. Wennerstrom, The colloidal domain where physics, chemistry,

biology and technology meet, VCH, New York, 1994.

2. D. Myers, Surfactant Science & Technology, second ed., VCH, New York,

1992.

3. Industrial applications of surfactants, the proceedings of a symposium

organized by the North West Region of the Industrial Division of the Royal

Society of Chemistry, University of Salford, 15-17th April 1986 (D.R. Karsa,

Ed.).

4. Industrial applications of surfactants II, the proceedings of a symposium

organized by the North West Region of the Industrial Division of the Royal

Society of Chemistry, University of Salford, 19th-20th April 1989 (D.R.

Karsa, Ed.).

5. Industrial applications of surfactants III, the proceedings of a symposium

organized by the North West Region of the Industrial Division of the Royal

Society of Chemistry. University of Salford, 16-18 September 1991 (D.R.

Karsa, Ed.).

6. P.M. Krugliakov, Hydrophile-lipophile balance of surfactants and solid

particles: physicochemical aspects and applications, Elsevier, Amsterdam,

Oxford, 2000.

7. J.N. Israelachvili, Intermolecular and surface forces, Academic Press, London,

1992.

8. R.G. Laughlin, Aqueous Phase Behavior of Surfactants, Academic Press, New

York, 1994.

9. K. Tsujii, Surface Activity: Principles, Phenomena & Applications, Academic

Press, London, 1998.

10. M.J. Rosen, Surfactants & Interfacial Phenomena, second ed., Wiley, New

York, 1989.

11. B. Jönsson, Surfactants and polymers in aqueous solution, Wiley, Chichester,

1998.

12. J.H. Clint, Surfactant aggregation, Blackie, Glasgow, 1992.

13. A.I. Rusanov, Micellization in Surfactant Solutions (Chemistry Reviews),

Taylor & Francis, 1998.

24

14. E. Ruckenstein, R. Nagarajan, Critical micelle concentration. Transition point

for micellar size distribution, J. Phys. Chem. 79 (1975) 2622-2626.

15. M. Smoluchowsky, Versuch einer mathematischen Theorie der

Koagulationskinetik kolloidaler Lösungen, Ztsch. Phys. Chem. 92 (1917) 129-

168.

16. V.M. Muller, Theory of reversible coagulation, Colloid. J. 58 (1996) 598-611.

17. P. J. Sams, E. Wyn-Jones, J. Rassing, New model describing kinetics of

micelle formation from chemical relaxation studies, Chemical Physics Letters,

13 (1972) 233-236.

18. J. Rassing, P.J. Sams, E. Wyn-Jones, Kinetics of micellization from ultrasonic

relaxation studies, J. Chem. Soc. Farad. II, 70 (1974) 1247-1258.

19. E.A.G. Aniansson, S.N. Wall, Kinetics of step-wise micelle association J.

Phys. Chem., 78 (1974) 1024-1030.

20. E.A.G. Aniansson, S.N. Wall, M. Almgren, H. Hoffman, I. Kielmann, W.

Ulbrich, R. Zana, J. Lang, C. Tondre, Theory of kinetics of micellar equilibria

and quantitative interpretation of chemical relaxation studies of micellar

solutions of ionic surfactants, J. Phys. Chem. 80 (1976) 905-922.

21. E. A. G. Aniansson, Dynamics and structure of micelles and other amphiphile

structures, J. Phys. Chem., 82 (1978) 2805-2808.

22. V.M. Muller, Theory of aggregative changes and stability of hydrophobic

colloids (in Russian). Theses, Doctor of Sciences, Moscow Institute of

Physical Chemistry (Russian Academy of Sciences), Moscow, 1982.

23. T. Telgmann, U. Kaatze, On the kinetics of the formation of small micelles. 1.

Broadband ultrasonic absorption spectrometry, J. Phys. Chem. B. 101 (1997)

7758-7765.

24. T. Telgmann, U. Kaatze, On the kinetics of the formation of small micelles. 2.

Extension of the model of stepwise association, J. Phys. Chem. B. 101 (1997)

7766-7772.

25. W.B. Russel, D.A. Saville, W.R. Scholwalter, Colloidal dispersions.

Cambridge University press, 1989.

25

FIGURE LEGENDS

Fig. 1

Disaggregation/aggregation kinetics according to Models A (A1 and A2), B (B1 and

B2), C (C1 and C2) and D (D1 and D2).

Model A

A1: disaggregation An+1→An+A1, n=2,3,4…

A2: aggregation An+A1→ An+1, n=1,2,3,4…

Model B B1: disaggregation Ai+j→Ai+Aj, i,j=1,2,3,4…

B2: aggregation Ai+Aj→Ai+j, i,j=1,2,3,4…

Model C C1: disaggregation An+1→An+A1, n=2,3,4…

C2: aggregation Ai+Aj→Ai+j, i,j=1,2,3,4…

Model D D1: disaggregation Ai+j→Ai+Aj, i,j=1,2,3,4…

D2: aggregation An+A1→ An+1, n=1,2,3,4…

Fig.2

Models A and B. Fractions of clusters 4,3,2,1, =kfk according to Eqs. (8)-(9) as

function of dimensionless concentration, α, under equilibrium conditions. No

restriction on concentration.

1 f1, single molecules,

2 f2, doublets,

3 f3, triplets,

4 f4 , quadruplets.

Fig.3

Model C. Fractions of clusters 3,2,1, =kfk according to Eqs. (A2.6) as functions

of dimensionless concentration, α, under equilibrium conditions at concentrations

below CMC (CMC=1 in dimensionless units).

1 f1, single molecules,

2 f2, doublets,

3 f3, triplets,

Concentration in the range 0<α<1.

26

Fig. 4 Model C. Transient behavior of dimensionless concentration of clusters, nfα , at different moments of dimensionless time, τ. Dimensionless concentration α=15 and the equilibrium number of molecules in micelles ℵ=200. 1 τ=1 2 τ=10 3 τ=18 4 τ=26 5 τ=38 6 τ=57.5

Fig. 5

Model C. Equilibrium distribution of concentration of clusters on dimensionless

concentration of surfactant molecules,α, at ℵ=200

1 α=1.5 (close to the dimensionless CMC)

2 α=15 (the same as in Fig 4)

3 α=150

Fig. 6

Model C. Averaged cluster size on dimensionless concentration. Inflection point is close to 1.5 and corresponds to the dimensionless CMC. ℵ=200

Fig. 7

Model C. Averaged cluster size on dimensionless concentration. Inflection point is close to 1.3 and corresponds to the dimensionless CMC. ℵ=50

Fig. 8

Model D. Dimensionless concentration of clusters on the dimensionless concentration, α.

1 α=1

2 α=10

3 α=100

4 α=1000

Fig. 9

Model D. Dependence of averaged cluster size on concentration, α.

27

Fig. 1

28

Fig. 2

29

Fig. 3

α

30

Fig. 4

31

Fig. 5 n

nfα

32

Fig. 6

33

Fig. 7

34

Fig. 8

αfn

n

4

3

2 1

35

Fig. 9